This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, then at least the answers might be of use not just for me.

To differentiate this from some questions already asked, let me clarify:

I am talking only about modern algebraic geometry, as in: everything that is better dealt with in terms of sheaves and schemes rather than varieties and curves. I know well enough that classical ("Italian") algebraic geometry has lots of applications; I am interested in knowing a reason to study (and a golden thread to follow in that) the kind of algebraic geometry that started with Serre, Leray, Grothendieck.

A "combinatorial/constructive algebraist" is a notion I cannot really formalize, but I mean an algebraist who is interested in actual computable things and their "fine structure" rather than topological abstracta and their "crude structure"; for example, actual polynomial identities rather than equality of zero-sets; actual isomorphisms instead of isomorphy; "for every point not on the zero-set of some particular ideal" rather than "for almost every point". The "combinatorial/constructive algebraist" (himself an abstraction) is fine with abstraction and formalism as long as he knows how to transform the abstract results into concrete equations and algorithms in case of need. He is not fine with nonconstructive existence results, although he is wary of declaring proofs unconstructive at first sight merely due to their formulation...

I believe I know of one example of this kind, a problem on matrix factorization solved using cohomology of sheaves somewhere on MathOverflow (any help with finding it is appreciated). There is also the interpretation of commutative Hopf algebras as coordinate Hopf algebras of affine schemes - but affine schemes are not really what I consider to be modern algebraic geometry; they correspond 1-to-1 to rings and are more frequently considered as functors than as locally ringed spaces in Hopf algebra theory. I would personally be more convinced by applications to invariant theory (viz., results from classical invariant theory proved with geometric methods) or the combinatorial kind of representation theory. I used to think that Swan's paper I linked in question 68071 is another application of scheme theory, but after understanding Seiler's proof it seems rather unnecessary to me.

It is not always easy to find out the underlying philosophy of a mathematician just by looking at some of its works, but these are very obvious cases: Gian-Carlo Rota, Doron Zeilberger, Donald Knuth, Henri Lombardi.
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darij grinbergOct 1 '11 at 21:23

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There were many reasons for introducing the tools of sheaves and cohomology, but an important one was to solve the classical problems in algebraic geometry that were not already solved. See Zariski's report on sheaves from the 1950s. (Zariski was a brilliant geometer from the era before sheaves, trained in the Italian school but not limited by its perspectives, and his report --- which is essentially a report on Serre's paper --- explains how the new sheaf-theoretic methods allow one to recover and generalize many results of the Italians and of Zariski himself.)
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EmertonOct 1 '11 at 23:28

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How about Weil's conjectures? At least one of them is a bit hard to prove without "modern" algebraic geometry.
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algoriOct 2 '11 at 4:34

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Is there some reason why this is not community wiki?
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Igor RivinOct 2 '11 at 15:17

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@Igor: I never make questions like that CW, because I believe that most answers to such a question do deserve the votes they get (the question itself, of course, doesn't; it is bound to be overrated). But if you want to make it CW, no problem.
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darij grinbergOct 2 '11 at 16:23

4 Answers
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A combinatorial motivation is the n! conjecture, whose proof by Haiman uses Hilbert schemes. An account of this work written by Haiman for the Current Developments in Mathematics conference in 2002 is at math.berkeley.edu/~mhaiman/ftp/cdm/cdm.pdf. Haiman emphasizes at the start of the paper that the main geometric results which had to be proved were motivated by combinatorial evidence. Around the time that Haiman first announced his results on the n! conjecture (before he moved to Berkeley) I had heard from other people that this conjecture motivated Haiman to learn modern algebraic geometry. Haiman's response to receiving the Moore prize in the AMS Notices April 2004, p. 432, more or less seems to confirm this, so it's analogous to the way that the Weil conjectures were a concrete open problem which motivated Grothendieck's work.

Very nice (I was aware of the conjecture but not of how it is proven), thanks!
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darij grinbergOct 2 '11 at 1:47

It is of course true that this is a nice concrete application of modern algebraic geometry. But people interested in explicit formulas have the right to expect more than these techniques give: we don't want an abstract reason for positivity as much as a manifestly positive formula.
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Stephen GriffethOct 2 '11 at 14:17

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SPG: By "positivity", which wasn't part of my answer, I assume you're referring to the Macdonald positivity conjecture, which was one of the corollaries of Haiman's proof of the n! conjecture. The positivity comes from an identification of the numbers in Macdonald's conjecture with character multiplicities, as character multiplicities are manifestly positive (well, manifestly nonnegative). See Conjecture 2.2.2 in Haiman's paper (Journal AMS 14 (2001), p. 945). I don't understand your complaint about not wanting an abstract reason for positivity. The positivity is not for an abstract reason.
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KConradOct 2 '11 at 16:27

KConrad: One could (and some combinatorialists do) complain that a proof that a certain vector space has dimension d should actually exhibit a basis with d elements. That is still open for the $n!$ problem. Also, for Macdonald positivity, one would like a combinatorial set that actually $q,t$-counts the coefficients of the Schur expansion (and ideally with it a way of creating a basis from this combinatorial set, and more ideally this basis respects the decomposition into $S_n$-irreps), and that question is still open (at least if one wants a direct definition of the set).
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Alexander WooOct 3 '11 at 8:04

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Alexander: If the desire for an alternate proof fitting a certain style successfully inspires someone, that's fair. Lots of second (or third) proofs arise for that reason. At the same time, even some elementary theorems only have proofs with an indirect character. For example, the only reason we know that the unit group $({\mathbf Z}/p{\mathbf Z})^\times$ is cyclic for every prime $p$ is by a non-explicit existence proof. If we refused to consider this theorem until such a time as someone found an explicit recipe for a generator, it may never be available for all $p$. :)
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KConradOct 3 '11 at 12:43

Positivity of Kazhdan--Lusztig polynomials (and all the other positivity results in Kazhdan--Lusztig theory in general).

Consider the Hecke algebra $H_n(q)$. It is a particular deformation of the group algebra of the symmetric group (or some other Coxeter group). As such, it has a basis $T_w$ indexed by permutations, and multiplication is given by
$$T_wT_{s_i}=T_{ws_i}$$
if $\ell(ws_i)=\ell(w)+1$, and
$$T_wT_{s_i}=qT_{ws_i}+(1-q)T_w$$
if $\ell(ws_i)=\ell(w)-1$.

Define an involution on $H_n(q)$ (usually called the bar involution) by $\overline{q}=q^{-1}$ and $\overline{T_w}=(T_{w^{-1}})^{-1}$. Kazhdan and Lusztig proved that there exists a unique basis $C^\prime_w$ such that

1) $\overline{C^\prime_w}=C^\prime_w$

2) If we write $C^\prime_w=\sum_x P_{x,w}(q)T_x$, then the degree of $P_{x,w}(q)$ is bounded above by $(\ell(w)-\ell(x)-1)/2$.

3) $P_{w,w}(q)=1$.

The polynomials $P_{x,w}(q)$ (and in particular their coefficient in the maximum degree they are allowed) turn out to give a very nice combinatorial way to construct representations of $S_n$ (or the Coxeter group in question), and a similar theory also constructs representations of finite groups of Lie type.

Now, the only way to prove that $P_{x,w}(q)$ have positive integer coefficients so far is to show that they are the Poincare polynomials for local intersection cohomology on Schubert varieties. Even better, one should interpret the Hecke algebra as a kind of Grothendieck group on the category of perverse sheaves on the flag variety. Springer in the early 1980s used this interpretation to show that, if one takes a product $C_vC_w$ and expands this product in the $C$ basis, the coefficients are all polynomials with positive integer coefficients. (The $C^\prime$ basis is a variant of the $C$ basis that is a little easier to write.)

(The best references I know are Humphrey's book on reflection groups and Coxeter groups and Bjorner and Brenti's book on Combinatorics of Coxeter groups, both of which have a chapter devoted to this subject.)

If you are interested in actual computations using modern algebraic geometry, there are plenty
to be had in Gromov-Witten theory and enumerative geometry. For example, Kontsevich's formula
counting rational plane curves is a famous example. The proof itself does not use any scheme theory, but it was based on the structure of a very delicate object called moduli space of stable maps, which could not be constructed without using schemes. Basically, counting problems
in enumerative geometry are usually transformed into intersection theory on moduli spaces
of the objects being counted. We want the moduli space to be compact so we can use the "invariant
of numbers" principle (example : two lines intersect at one point in projective plane, but not necessarily so in
affine line). Compactifying the moduli space meaning you allow your objects to have limits, thus even if your objects of interested are smooth varieties, their limits may be deformed and have unreduced structures (read : schemes and sheaves). For example, a conic may be denegrate to a double line which only makes sense as a scheme.
You can read a nice exposition of Kontsevich's formula in : http://arxiv.org/abs/alg-geom/9608011.

As another example, in http://arxiv.org/abs/alg-geom/9612004 , Getzler computed the intersection pairing matrix of $\mathcal M_{1,4}$ to obtain relation between cycles and then use it to compute, for example, the number of elliptic curves of degree $5$ passing through $20$ lines in $\mathbb P^3$ to be $2,583,319,387,968$, among other thing.

Just from glancing through the paper, it seems to me that it uses only affine schemes, which makes me wonder whether it couldn't as well be worded in terms of Hopf algebras. Have I missed some non-affine things?
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darij grinbergOct 2 '11 at 16:29

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@Darij: Even if only affine schemes are used, it is worthwile to mention that there are group theoretic questions which can be attacked by thinking in terms of modern algebraic geometry.
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Martin BrandenburgOct 2 '11 at 20:43

Given that they are looking at quasi-fixed points of polynomial dynamical systems over finite fields, I don't think it is natural to change the language to commutative rings. I haven't read the paper in awhile, but I think schemes, maybe affine, over Z are used. This is in a sense modern algebraic geometry since classical algebraic Italian style algebraic geometry was over fields and considered only reduced affine schemes.
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Benjamin SteinbergOct 3 '11 at 1:59

They also mention how free groups can be replaced by f.g. linear groups by using a result of Hrushovski that does seem to use modern algebraic geometry.
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Benjamin SteinbergOct 3 '11 at 2:14

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Our paper with Borisov is using commutative algebra and affine schemes. We tried to use algebraic geometry (and arbitrary schemes) but we did not succeed because there is no satisfactory intersection theory in positive characteristic. In the paper, we indicate precisely the intersection theory facts we needed. That and much more has been done by Hrushovski in his paper about Frobenius in characteristic $0$. The main results of Hrushovski are geometric but the methods are from logic and model theory.
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Mark SapirOct 5 '11 at 1:10